Title: An example of Hilbert's Irreducibility Theorem
Speaker: Michael Filaseta
Abstract: We give a brief overview of Hilbert's Irreducibility Theorem. The main purpose of the talk is to provide an informative example that will help with understanding how Hilbert's Irreducibility Theorem is a consequence of Siegel's Theorem. For the example, we will use the polynomial f(x,y) = x^4 + (y+8) x - y^2 - y, and show that this polynomial is irreducible as a polynomial in x over the integers for an integer y if and only if y is not in the set { -96, -39, -3, -1, 0, 3, 7, 32, 105 }.
Title: On the Galois group over Q of a truncated binomial expansion
Speaker: Michael Filaseta
Abstract: This will be a somewhat general talk about Galois groups and methods for computing the Galois group of a polynomial. We then turn to an application of these methods involving joint work with Richard Moy on the Galois group over the rationals associated with a truncation of the binomial expansion (1+x)^n.
There was no seminar this week.
Title: An integer partition inequality of Bessenrodt-Ono
Speaker: Stephen Gagola
Abstract: Here we give a combinatorial proof of an inequality that was first proven by Christine Bessenrodt and Ken Ono. C. Bessenrodt and K. Ono proved that the number of partitions of n, say p(n), satisfies p(a)p(b) > p(a + b) for a, b > 1 and a + b > 9 by using a result of Lehmer and asked whether a combinatorial proof exists. Here we prove the inequality combinatorially and show that the proof can also be extended to prove the analogous inequality for other partition functions including k-regular partitions with k >= 2. For 2 <=k <= 6, these inequalities were first proven to hold for k-regular partitions by Olivia Beckwith and Christine Bessenrodt using similar methods to the p(n) case.
Title: Lattice points on smooth curves
Speaker: Ognian Trifonov
Abstract: We go over work of Swinnerton-Dyer where he essentially shows that for every epsilon > 0 there exists a constant c(epsilon ) > 0 such that for every strictly convex curve C with continuous third derivative "with a sensible bound", there are at most c(\epsilon ) l^{3/5 + epsilon} lattice points on C, where l is the length of C.
Title: Three perhaps doable Number Theory problems
Speaker: Michael Filaseta
Abstract: I will present three (mini) research problems that are perhaps doable which graduate students can feel free to look at if they want to. I will give brief explanations as to why I think they are doable. The first has to do with a p-adic analog for the Prouhet-Tarry-Escott problem and boils down to a question in modulo arithmetic. The second has to do with zeroes of polynomials in the unit disk. The third is a Galois theory question.
Title: An Introduction to Arithmetic Statistics
Speaker: Frank Thorne
Abstract: Arithmetic statistics" is all about counting arithmetic objects: number fields, class groups, elliptic curves, and so on. I'll give an overview of the subject and some interesting theorems that have been proved recently.
Title: Malle's conjecture
Speaker: Harsh Mehta
Abstract: We look at an overview of Malle's conjecture, some applications, and discuss some problems that are of interest to me.